While all the Hobbits I can think of have a combination of first- and lastname (e.g., Bilbo Baggins, Peregrin Took, the Sackville-Baggins's,...), I can't think of any character* of another race that has a lastname:

Men seem to like to add a reference to their ancestors (e.g., Aragorn, Arathorn's son or Isildur's heir)

Dwarves seem to like descriptive or honorary names (e.g., Thorin Oakenshield, Dain Ironfoot). They are not lastnames though, since they do not get passed on to descendants.

Elves don't have a need for lastnames, since elvish names are unique. Some nevertheless seem to have an additive (e.g., Legolas Greenleaf).

@Richard: I only agree with Barliman Butterbur. The rest do not have last names, they are descriptive additives (in your examples, Elrond and Arwen do not share a name; if they had lastnames, both father and daughter would have the same).
– mortApr 5 '14 at 16:00

@Richard: Brothers would usually share the same last name, formed by adding "sen" to their father's first name. But a father and son would not usually have the same last name (unless the father and grandfather happen to share the same first name, which I think would not be common).
– Keith ThompsonApr 5 '14 at 20:47

1 Answer
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There's no writing that I can find in either HoME or Letters bearing on this question. A few mentions of family names, general discussion of the names themselves, but nothing relating to why Hobbits (and, presumably, Bree-folk) have this particular distinction.

Perhaps the closest is Letter 25, from which I'll quote:

The dwarf-names, and the wizard's, are from the Elder Edda. The hobbit names from Obvious Sources proper to their kind. The full list of their wealthier families is: Baggins, Boffin, Bolger, Bracegirdle, Brandybuck, Burrowes, Chubb, Grubb, Hornblower, Proudfoot, Sackville, and Took. The dragon bears as name – a pseudonym – the past tense of the primitive Germanic verb Smugan, to squeeze through a hole: a low philological jest. The rest of the names are of the Ancient and Elvish World, and have not been modernised.

It's probably worth considering the narrative structure of both the Hobbit and LotR. In both cases, Hobbits are intended to be a "touchstone" for the reader, and both books start with the relative familiarity of the Shire. We view the main action through their eyes, and as the story unfolds it passes from comfortable and familiar surroundings into the world of myth. Tolkien himself was aware of this function of Hobbits, as he notes in Letter 182 (discussing the Silmarillion):

Though I do not think it would have the appeal of the L. R. – no hobbits ! Full of mythology, and elvishness, and all that 'heigh stile' (as Chaucer might say), which has been so little to the taste of many reviewers.

Christopher Tolkien also notes the same in his foreword to the Lost Tales: "in The Silmarillion (there is) no 'mediation' of the kind provided by the Hobbits".

It's worth noting that Bree-folk were mentioned in the comments to the question, and I touched on them above; we should remember that they also have family names. Barliman Butterbur, Bill Ferny, Harry Goatleaf. As Tolkien notes in FotR (At the Sign of the Prancing Pony):

The Men of Bree seemed all to have rather botanical (and to the Shire-folk rather odd) names, like Rushlight, Goatleaf, Heathertoes, Appledore, Thistlewool and Ferny (not to mention Butterbur).

This supports the thesis that names also reflect the passage from familiar territory to the mythical world. Here we see that in Bree we have the use of family names for familiarity, but the family names used seem odd to the Hobbits. They're leaving their familiar territory, and soon enough they'll be out in the Wild. It's interesting to observe that Aragorn (in his "Strider" guise) - something of a pariah in Bree - has no family name.

In the light of this, it's probably not surprising that the Shire and it's environs were where family names were used, thereby providing another element of initial familiarity.

Indeed. For all intents and purposes, the Shire is an idealised piece of 19th-century southern England dropped into the ancient/mythic landscape of Middle Earth. It includes things like tea, tobacco (er, pipe-weed) and potatoes which were unknown in England until fairly recent times. So it's not surprising that the hobbits' names feel a bit anachronistic as well.
– Royal Canadian BanditApr 5 '14 at 18:18